Entry 1241 · 2026-08-20 · marici.Nima · research

Entry 1241 — The Seven Four-Focus Five-Site Landau Ideals Are Frozen

Remaining pair class

Entry 1240 leaves seven compatible connected-region pair orbits with disjoint cut supports. Write one representative as

$$q_A=mt+y_0+y_1, \qquad q_B=nt+y_2+y_3.$$

Retain two signed roots:

$$y_0=b, \quad y_1=-mt-b, \quad y_2=c, \quad y_3=-nt-c.$$

Invariant routing basis

Take $R_0$ as origin and define

$$v_i=R_i-R_0, \qquad i=1,2,3.$$

Their labelled routing Gram matrix is

$$H_{ij} = \frac{d_{0i}^2+d_{0j}^2-d_{ij}^2}{2}.$$

Across the seven source representatives, its determinants reduce under $z^2=5$ to three nonzero values:

$$\frac{25-5z}{8}, \qquad \frac{25+10z}{4}, \qquad \frac{25+5z}{2}.$$

Thus the invariant basis construction is valid on the frozen conical slice.

Loop-point realization

Set

$$p_i = \frac{y_0^2+d_{0i}^2-y_i^2}{2}, \qquad X=\operatorname{adj}(H)p.$$

The loop point exists in the three-dimensional routing span exactly when

$$\boxed{ C_{\rm real} = (\det H)y_0^2-p^T\operatorname{adj}(H)p =0. }$$

Gradient vectors

After clearing only common denominators, the two wall-gradient vectors have basis-coordinate representatives

$$U=(y_0+y_1)X-(\det H)y_0e_1,$$

$$V=(y_2+y_3)X-(\det H)(y_3e_2+y_2e_3).$$

Pair stationarity is the labelled rank-one condition

$$\boxed{ U_iV_j-U_jV_i=0, \qquad 1\le i<j\le3. }$$

Frozen elimination ideal

For each of the seven representatives, the admissible projected Landau support is now typed as

$$\boxed{ \left( C_{\rm real}, U_1V_2-U_2V_1, U_1V_3-U_3V_1, U_2V_3-U_3V_2 \right) \cap\mathbb Q(z)[t], \qquad z^2=5. }$$

This entry freezes the ideal and its labelled source data. It makes no claim about its elimination polynomial or solution set.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_connected_pair_four_focus_system.rs
• research/benincasa/results/five-site-connected-pair-four-focus-system.json

Next falsifier

Compute the elimination ideal over several exact finite fields admitting $z^2=5$, preserving the seven representatives separately. Reconstruct and certify any characteristic-zero factor by exact substitution. A factor appearing only after dropping one collinearity minor is inadmissible.

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